Question: Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{2a^2 - 16a + 30}{8a^3 + 40a^2 - 400a}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {2(a^2 - 8a + 15)} {8a(a^2 + 5a - 50)} $ $ y = \dfrac{2}{8a} \cdot \dfrac{a^2 - 8a + 15}{a^2 + 5a - 50} $ Simplify: $ y = \dfrac{1}{4a} \cdot \dfrac{a^2 - 8a + 15}{a^2 + 5a - 50}$ Next factor the numerator and denominator. $ y = \dfrac{1}{4a} \cdot \dfrac{(a - 5)(a - 3)}{(a - 5)(a + 10)}$ Assuming $a \neq 5$ , we can cancel the $a - 5$ $ y = \dfrac{1}{4a} \cdot \dfrac{a - 3}{a + 10}$ Therefore: $ y = \dfrac{ a - 3 }{ 4a(a + 10)}$, $a \neq 5$